PUBLICATIONS OF DORON ZEILBERGER

Last Update: May 4, 2005 This list, as well as most of the papers published after 1990, are available at http://www.math.rutgers.edu/~zeilberg.

1. (With D. Nicholson, P. Rabinowitz and N. Richter) On the Error in the Numerical Integration of Chebyshev Polynomials, Math. Computation 25, 79-86 (1971).

2. Uniqueness Theorems for Harmonic Functions of Exponential Growth, Proc. Amer. Math. Soc. 61, 335-340 (1976).

3. Binary Operations in the set of solutions of a Partial Difference Equation, Proc. Amer. Math. Soc. 62, 242-244 (1977).

4. A new approach to the theory of Discrete Analytic Functions, J. Math. Anal. Appl. 57, 350-367 (1977).

5. (With H.Dym) Further properties of Discrete Analytic Functions, J. Math. Anal. Appl. 58, 405-418 (1977).

6. A new basis for Discrete Analytic Functions, J. Australian Math. Soc. 23 (series A), 95-104 (1977).

7. Discrete Analytic Functions of exponential growth, Trans. Amer. Math. Soc. 226, 181-189 (1977).

8. A discrete analog of the Paley-Wiener theorem in a half plane, J. Australian Math. Soc. (Series A) 23, 376-378 (1978).

9. A new proof of Ehrenpreis’s semi-local Quotient Structure Theorem, Amer. J. Math. 100, 1317-1332 (1978).

10. The Pompeiu problem for discrete space, Proc. Natl. Acad. Sci. 75, 3555-3556 (1978).

11. Solutions of exponential growth to systems of partial differential equations, J. Diff. Eq. 31, 287-295 (1979).

12. The algebra of linear partial difference operators and its applications, SIAM J. Math. Anal. 11, 919-934 (1980).

13. Some comments on Rota’s umbral calculus, J. Math. Anal. Appl. 74, 456-463 (1980).

14. A lattice walk approach to the q-counting of multiset permutations, J. Math. Anal. Appl. 74, 192-199 (1980).

15. Partial difference equations in m_{1} ≥ ... ≥ m_{n} ≥ 0 and their applications to combinatorics, Discrete Math 31, 65-77 (1980).

16. Enumerating words by their number of mistakes, Discrete Math 34, 89-91 (1981).

17. All binomial identities are verifiable, Proc. Natl. Acad. Sci. 78, 4000 (1981).

18. Sister Celine’s technique and its generalizations, J. Math. Anal. Appl. 85, 114-145 (1982).

19. (With D. Franzblau) A bijection proof of the hook length formula, J. Algorithms 3, 317-343 (1982).

20. (With D. Bressoud) A short Rogers-Ramanujan bijection, Discrete Math. 38, 313-315 (1982).

21. A combinatorial proof of Dyson’s conjecture, Discrete Math. 41, 317-321 (1982).

22. (With S. R. Caplan) T. L. Hill’s graphical method for solving linear equations, Advances in Appl. Math. 3, 377-383 (1982).

23. (With L. Shapiro) A Markov chain occurring in Enzyme Kinetics, J. Math. Biology 15, 351-357 (1982).

24. (With J. Gillis) A direct combinatorial proof of a positivity result, European J. Comb. 4, 221-223 (1983).

25. Andre’s reflection proof generalized to the many-candidate ballot problem, Discrete Math 44, 325-326 (1983).

26. (With J. Gillis and B. Reznick) Elementary methods in positivity theory, SIAM J. Math. Anal. 14, 396-398 (1983).

27. A combinatorial proof of Newton’s identities, Discrete Math. 49, 319 (1984).

28. A short hook-length bijection inspired by the Greene-Nijenhuis-Wilf proof, Discrete Math. 51, 101-108 (1984).

29. Garsia and Milne’s proof of the inclusion-exclusion principle, Discrete Math. 51, 109-110 (1984).

30. (With D. Foata) Weighted derangements and Laguerre polynomials, Actes Sémin. Lotharingien de combinatoire 8, 20-29 (1984).

31. (With E. A. Bender) Some asymptotic bijections, J. Comb. Th. (Ser A) 38, 96-98 (1985).

32. (With D. Bressoud) A proof of Andrews’ q-Dyson conjecture, Discrete Math. 54, 201-224 (1985).

33. (With J. Wimp) Resurrecting the asymptotics of linear recurrences, J. Math. Anal. Appl. 111, 162-177 (1985).

34. A combinatorial approach to matrix algebra, Discrete Math. 56, 61-72 (1985).

35. (With D. Bressoud) Bijecting Euler’s partitions recurrence, Amer. Math. Monthly 92, 55-56 (1985).

36. (With M. Werman) Bijecting Cassini’s Fibonacci identity, Discrete Math. 58, 109 (1986).

37. Toward a combinatorial proof of the Jacobian conjecture?, Proc. of the ”Colloque de combinatoire énumérative”, G.Labelle and P.Leroux, editors, Lecture Notes in Math. 1234, 370-380, Springer-Verlag, Berlin (1987).

38. A proof of the G_{2} case of Macdonald’s root system-Dyson conjecture, SIAM J. Math. Anal. 18, 880-883 (1987).

39. (With D. Bressoud) Generalized Rogers-Ramanujan identities, Advances in Math. 78, 42-75 (1989).

40. A q-Foata proof of the q-Saalschütz identity, European J. Comb. 8, 461-463 (1987).

41. Enumerating totally clean words, Discrete Math. 64, 313-315 (1987).

42. One line proofs of the unimodality of The Gaussian polynomials G(n, k), for k = 3, 4, Ars Comb., 24, 165-166 (1987).

43. A bijection from ordered trees to binary trees that sends the pruning order to the Strahler number, Discrete Math. 82, 89-92 (1990).

44. (With J. Gillis and J. Jedwab) A combinatorial interpretation of the integral of the products of Legendre polynomials, SIAM J. Math. Anal. 19, 1455-1461 (1988).

45. A unified approach to Macdonald’s root-system conjectures, SIAM J. Math. Anal. 19, 987-1013 (1988).

46. A Stembridge-Stanton style proof of the Habsieger-Kadell q-Morris identity, Discrete Math. 79, 313-322 (1989/90).

47. (With D. Foata) Laguerre polynomials, weighted derangements, and positivity, SIAM J. Discrete Math. 1, 425-433 (1988).

48. (With D. Foata) Linearization coefficients for the Jacobi polynomials, Actes Séminaire Lotharingien 16, 73-86, Publ. I.R.M.A., Strasburg (1988).

49. Six Etudes in generating functions, Intern. J. Computer Math. 29, 201-215 (1989).

50. A combinatorial problem that arose in biophysics, Fibonacci Quarterly 27, 372 (1989).

51. On a conjecture of R.J.Simpson about exact covering sequences, Amer. Math. Monthly 96, 243 (1989).

52. Kathy O’hara’s constructive proof of the unimodality of the Gaussian polynomials, Amer. Math. Monthly 96, 590-602 (1989).

53. One line proofs of the unimodality of the Gaussian polynomials G(n, k) for k < 20, in: D. Stanton, ed., ”q-Series and Partitions”, (Proc., IMA, March 1988), IMA series 18, Springer, 35-44 (1989).

54. (With D. Stanton) The Odlyzko conjecture and O’Hara’s unimodality proof, Proc. Amer. Math. Soc. 107, 39-42 (1989).

55. Identities, in: D. Stanton, ed., ”q-Series and Partitions”, (Proc., IMA, March 1988), IMA series 18, 67-75 (1989).

56. A Holonomic systems approach to special functions identities, J. of Computational and Applied Math. 32, 321-368 (1990).

57. (With S. B. Ekhad) A 21^{st} century proof of Dougall’s hypergeometric identity, J. Math. Anal. Appl. 147, 610-611 (1990).

58. A Fast Algorithm for proving terminating hypergeometric identities, Discrete Math 80, 207-211, (1990).

59. (With H. S. Wilf) Rational functions certify combinatorial identities, J. Amer. Math. Soc. 3, 147-158 (1990).

60. (With J. Wimp) How likely is Polya’s drunkard to stay in x ≥ y ≥ z ?, J. Statistical Physics 57, 1129-1135 (1989).

61. (With G. Almkvist) The method of differentiating under the integral sign, J. Symbolic Computation 10, 571-591 (1990).

62. The method of creative telescoping, J. Symbolic Computation 11, 195-204 (1991).

63. (With D. Foata) Denert’s permutation statistic is indeed Euler-Mahonian, Studies in Applied Math 83, 31-59 (1990).

64. (With D. Foata) Multibasic Eulerian polynomials, Trans. Amer. Math. Soc. 328, 843-862 (1991).

65. (With R. J. Simpson) Necessary conditions for distinct covering systems with square-free moduli, Acta Arithmetica 59, 59-70 (1991).

66. (With H. S. Wilf) Towards computerized proofs of identities, Bulletin of the Amer. Math. Soc.

23, 77-83 (1990).

67. Gauss’s _{2}F_{1}(1) cannot be generalized to _{2}F_{1}(x), J. Comp. Appl. Math. 39, 379-382 (1992).

68. (With I. Gessel) Random Walk in a Weyl chamber, Proc. Amer. Math. Soc. 115, 27-31 (1992).

69. Closed Form (pun intended!), in: ”Special volume in memory of Emil Grosswald”, M. Knopp and M. Sheingorn, eds., Contemporary Mathematics 143 579-607, AMS, Providence (1993).

70. Gert Almkvist’s generalization of a mistake of Bourbaki, in: ”Special volume in memory of Emil Grosswald”, M. Knopp and M. Sheingorn, eds., Contemporary Mathematics 143 609-612, AMS, Providence (1993).

71. Three recitations on Holonomic Systems and Hypergeometric Series, Proceeding of the Séminaire Lotharingien de combinatoire 24,5-37, IRMA, Strasbourg. (1993) (To be reprinted in Journal Symbolic Computation.)

72. A constant term identity featuring the ubiquitous (and mysterious) Andrews-Mills-Robbins-Rumsey numbers {1, 2, 7, 42, 429, ...}, J. Combinatorial Theory (ser. A) 66, 17-27 (1994).

73. A proof of Julian West’s conjecture that the number of 2-stack-sortable permutations of length n is 2(3n)!/((2n + 1)!(n + 1)!), Discrete Math. 102, 85-93 (1992).

74. Identities in search of identity, J. Theoretical Computer Science 117, 23-38 (1993).

75. (With Jane Friedman) A generalization of Odlyzko’s conjecture: the coefficients of (1−q)^{j}/((1− q^{2n})...(1 − q^{2n+2j})) alternate in sign, Proc. Amer. Math. Soc. 118, 1013 (1993).

76. (With H.S. Wilf) An algorithmic proof theory for hypergeometric (ordinary and ”q”) multisum/integral identities, Invent. Math. 108, 575-633 (1992).

77. (With S.B. Ekhad) A one-line WZ proof of a formula of Ramanujan for π, in: “Geometry, Analysis, and Mechanics” (Volume to honor Archimedes’s 2281st birthday), J. M. Rassias, ed., 107-108. World Scientific, Singapore (1994). 78.(with Jane Friedman and Ira Gessel) Talmudic lattice path counting, J. Combin. Theory Ser. A 68, 215-217 (1994).

79. Proof of q-analog of a constant term identity conjectured by Forrester, J. Combinatorial Theory 66, 311-312 (1994).

80. (With H. S. Wilf) Rational function certification of hypergeometric multi-integral/sum/”q” identities, Bulletin of the Amer. Math. Soc. 27, 148-153 (1992).

81. (With G.E. Andrews and S.B. Ekhad), A short proof of Jacobi’s formula for the number of representations of an integer as a sum of four squares, Amer. Math. Monthly, 100, 274-276 (1993).

82. Towards a WZ proof of Mehta’s integral, SIAM J. Math. Anal. 25 , 812-814 (1994).

83. (With S. B. Ekhad) A short and elementary, “formal calculus” proof of the Bieberbach conjecture (after L. Weinstein), Contemporary Math 178 113-115, (1995).

84. (With D. Foata) Combinatorial Proofs of Cappelli’s and Turnbull’s Identities from Classical Invariant Theory, Electronic J. of Combinatorics(http://www.combinatorics.org) , 1, R1 ( 7 pages) (1994).

85. Proof of the alternating sign matrix conjecture, Elect. J. Combinatorics (http://www.combinatorics.org) 3(2) [Foata Festschrift] R13, (50 pages) (1996).

86. Theorems for a price: Tomorrow’s semi-rigorous mathematical culture, Notices of the Amer. Math. Soc. 40 # 8, 978-981 (Oct. 1993). Reprinted: Math. Intell. 16, no. 4, 11-14 (Fall 1994).

87. Chu’s 1303 identity implies Bombieri’s 1990 norm-inequality [Via an identity of Beauzamy and Dégot], Amer. Math. Monthly 101, 894-895 (1994).

88. How Joe Gillis discovered Combinatorial Special Function Theory, Math. Intell. 17, no. 2, 65-66 (Spring 1995).

89. (With L. Ehrenpreis) Two EZ proofs of sin^{2} z + cos^{2} z = 1, Amer Math. Monthly 101, 691 (1994).

90. (With C. Orr), A computer algebra approach to the discrete Dirichlet problem, J. Symbolic Comput. 18, 87-90 (1994).

91. On an identity of Daubechies, Amer. Math. Monthly 100, 487 (1993).

92. The J.C.P. Miller Recurrence for Exponentiating a polynomial and its q-Analog, J. Difference Eqs. and Appls. 1, 57-60 (1995).

93. Proof of the refined alternating sign matrix conjecture, New York J. of Math. (elec.), (http://nyjm.albany.edu/) 2, 59-68 (1996).

94. (With D. Foata), The Graphical Major Index, J. Comp. Appl. Math., [specal issue on q-series], J. Comput. Applied Math (special issue on q-series) 68 79-101 (1996).

95. (With J. Noonan), Counting Permutations with a prescribed number of ”forbidden” patterns, Advances in Applied Math. 17, 381-407 (1996).

96. Reverand Charles to the aid of Major Percy and Fields-Medalist Enrico, Amer. Math. Monthly 103, 501-502 (1996).

97. The method of undetermined generalization and specialization illustrated with Fred Galvin’s amazing proof of the Dinitz conjecture, Amer. Math. Monthly 103, 233-240 (1996).

98. An explicit formula for the number of solutions of X^{2} = 0 in triangular matrices over GF(q), Elect. J. Comb(http://www.combinatorics.org). 3(1), R3 (3 pages)(1996).

99. Self-Avoiding Walks, the language of science, and Fibonacci numbers, J. Stat. Planning and Inference 54, 135-138 (1996).

100. If A_{n} has 6n dyes in a box, with which he has to fling at least n sixes, then A_{n} has an easier task than A_{n+1}, at Eaven Luck, Amer. Math. Monthly 103, 265 (1996).

101. (With S. B. Ekhad)Curing the Andrews Syndrome, J. of Difference Equations and Applications 4, 299-310 (1997).

102. (With T. Amdeberhan) Hypergeometric Series Acceleration via the WZ method Elect. J. of Combinatorics (http://www.combinatorics.org), 4(2) [Wilf Festschrifft volume], R3 (4 pages) (1997).

103. (With T. Amdeberhan) q-Apery Irrationality Proofs by q-WZ Pairs, Adv. Appl. Math. 20, 275-283, (1998).

104. (With D. Foata) A classic proof of a recurrence for a very classical sequence, J. Combin. Theory Ser. A 80, 380-384 (1997).

105. (With D. Foata) Combinatorial Proofs of Bass’s Evaluations of the Ihara-Selberg Zeta function of a Graph, Trans. Amer. Math. Soc., 351, 2257–2274 (1999).

106. The Abstract Lace Expansion, Advances in Applied Mathematics, 19, 355-359 (1997).

107. (With I. Nemes, M. Petkovsek, and H. S. Wilf) How to do Monthly Problems on your computer, Amer. Math. Monthly 104, 505-519 (1997).

108. Dodgson’s Determinant-Evaluation Rule Proved by TWO-TIMING MEN and WOMEN, Elect. J. of Combinatorics (http://www.combinatorics.org) 4(2), [Wilf Festschrift volume], R22 (2 pages) (1997).

109. A comparison of two methods for random labellings of balls by vectors of integers, Advances in Combinatorial Methods and Applications to Probability and Statistics, N. Balakrishnan, ed., Birkhauser, 1997 (Mohanty Festschrift).

110.(With S. B. Ekhad) Proof of Conway’s Lost Cosmological Theorem Elect. Res. Announcements of the AMS 3, 78-82 (1997).

111. Math Bite: Proof of an Empirical Observation Made by Amos Oz’s character, Math. Magazine 70 , 291 (1997).

112. How Much Should a Nineteenth-Century French Bastard Inherit, J. Difference Eq. Appl. 3, 385-388 (1998).

113. (With C. Krattenthaler) Proof of a Determinant Evaluation Conjectured by Bombieri, Hunt, and van der Poorten, New York J. of Math. (elec.) (http://nyjm.albany.edu/) 3, 54-102 (1997).

114. Automated Counting of LEGO Towers, J. Difference Eq. Appl., 5, 323-333, (1999).

115. (With J. Noonan) The Goulden-Jackson Cluster Method: Extensions, Applications, and Implementations, J. Difference Eq. Appl. 5, 355-377, (1999).

116. (With Scott Ahlgren, Shalosh B. Ekhad, Ken Ono) A Binomial Coefficient Identity Associated to a Conjecture of Beukers, Electronic Journal of Combinatorics (http://www.combinatorics.org) 5, R10 (1 page) (1998).

117. (With Aaron Robertson) A 2-Coloring of [1,N] Can Have (1/22)N^{2} + O(N) Monochromatic Schur Triples, But Not Less!, Electronic Journal of Combinatorics (http://www.combinatorics.org) 5 R19, (5 pages) (1998).

118. (With Shalosh B. Ekhad) There Are More Than 2**(n/17) n-Lettered Ternary Square-Free Words, J. Integer Sequences (elec.) (http://www.research.att.com/ njas/jit/), 98.1.9 (3 pages) (1998).

119. Enumeration Schemes, and More Importantly, Their Automatic Generation, Annals of Combinatorics 2, 185-195 (1998).

120. Proof Of A Conjecture Of Chan, Robbins, and Yuen, ETNA, (Elec. Trans, of Numerical Analysis) (elec.) (http://etna.mcs.kent.edu/) 9, 147-148, (1999).

121. (With Aaron Robertson and Herb Wilf) Patterns and Fractions, Elec. J. Combinatorics, 6,

(http://www.combinatorics.org) R38, (4 pages) (1999).

122. (With S. B. Ekhad) PLANE GEOMETRY: An Elementary School Textbook (ca. 2050), Mathematical Intelligencer 21(3), 64-70, (1999).

123. Symbol-Crunching with the Transfer-Matrix Method in Order to Count Skinny Physical Creatures, INTEGERS (http://www.integers-ejcnt.org), 0 A9 (29 pages) (2000).

124. (With A. Edlin) The Goulden-Jacskon Cluster Method For Cyclic Words, Advances in Applied Mathematics 25, 228-232, (2000).

125. The Umbral Transfer-Matrix Method: I. Foundations, J. Comb. Theory Ser. A 91, 451-463, (Rota memorial issue) (2000).

126. How Berger, Felzenbaum, and Fraenkel Revolutionized COVERING SYSTEMS The Same Way that George Boole Revolutionized LOGIC, Elect. J. Combinatorics 8(2) (2001) (special issue in honor of Aviezri Fraenkel), A1 (http://www.combinatorics.org) , (9 pages).

127. (With D. Foata) Babson-Steingrimsson Statistics Are Indeed Mahonian (and Sometimes Even Euler-Mahonian), Adv. Appl. Math. 27 (2001), 390-404

128. Three-Rowed CHOMP, Adv. Appl. Math. 26 (2001), 168-179.

129. The Umbral Transfer-Matrix Method. III. Counting Animals, New York J of Mathematics 7(2001), 223-231.

130. (With T. Amdeberhan) Determinants Through The Looking Glass, Adv. Appl. Math. 27 (2001), 225-230

131. The Umbral Transfer-Matrix Method. IV. Counting Self-Avoiding Polygons and Walks, Elec. J. Comb. 8(1)(2001), (22 pages) R28.

132. The Umbral Transfer-Matrix Method. V. The Goulden-Jackson Cluster Method for Infinitely Many Mistakes, INTEGERS, 2 (2002), (10 pages), A5 .

133. (With D. Foata) The Collector’s Brotherhood Problem Using the Newman-Shepp Symbolic Method, Algebra Universalis (special Rota memorial issue), Algebra Universalis (special Rota memorial issue),49 (2003), 387-395.

134. “Real” Analysis is a Degenerate Case of Discrete Analysis, Appeared in: ”New Progress in Difference Equations”(Proc. ICDEA 2001), edited by Bernd Aulbach, Saber Elaydi, and Gerry Ladas, and publisher by Taylor & Francis, London, 2004.

135. Computerized Deconstruction, Adv. Applied Math. 30 (2003), 633-654.

136. (With Aaron Robertson, Dan Saracino) Refined Restricted Permutations, Annals of Combinatorics. 6 (2003), 427-444.

137. Lieber Opa Paul, Ich Bin Auch Ein Experimental Scientist, Adv. Appl. Math. 30 (2003), 633-654.

138. Towards a SymbolicComputational Philosophy (and Methodology!) for Mathematics, Proc. of the BuchbergerFest, (Peter Paule, ed.), Springer, to appear.

139. (With Stavros Garoufalidis and Thang TQ Le) The Quantum MacMahon Master Theorem ,submitted.

140. Chomp, Recurrences, and Chaos, to Appear in J. Difference Equations and its Applications, J. Difference Equations and its Applications 10 (2004), 1281-1293. (special issue in honor of Saber Elaydi).

141. (With Xinyu Sun) On Fraenkel’s N-Heap Wythoff Conjecture, Annals of Combinatorics 8 (2004). 225-238.

142. (With Mohamud Mohammed) The Markov-WZ Method Elec J. Combinatorics 11(2004), R53. (14 pages). 143.Symbolic Moment Calculus I.: Foundations and Permutation Pattern Statistics, Appeared in Annals of Combinatorics 8 (2004), 369-378.

144. Dave Robbins’s Art of Guessing, Adv. Appl. Math. 34 (2005), 939-954. [special issue in memory of David Robbins].

145. (With Mohamud Mohammed) Sharp Upper Bounds for the Orders of the Recurrences Outputted by the Zeilberger and q-Zeilberger Algorithms, J. Symbolic Computation 39 (2005), 201-207.

146. Enumerative and Algebraic Combinatorics, to appear in: ‘Princeton Companion of Mathematics’, T.W. Gowers, ed.

147. (With Arthur Benjamin) Pythagorean Primes and Palindromic Continued Fractions, submitted.

148. (With Andrew V. Sills) Disturbing the Dyson Conjecture (in a GOOD Way), submitted.

149. DECONSTRUCTING the ZEILBERGER algorithm, to appear in J. of Difference Equations and Applications.

150. (With Mohamud Mohammed) Multi-Variable Zeilberger and Almkvist-Zeilberger Algorithms and the Sharpening of Wilf-Zeilberger Theory, submitted. BOOKS B1. (With M. Petkovsek and H. S. Wilf) A=B, AK Peters, Wellesley, (1996). B2. Shalosh B. Ekhad’s 2050 Plane Geometry Text, (webbook,

http://www.math.rutgers.edu/~zeilberg/PG.html)

B3. Shalosh B. Ekhad’s Computer-Generated Alphametics, (webbook,

http://www.math.rutgers.edu/~zeilberg/hans/hans.html)

BOOK REVIEWS

1. Review of ”The Umbral Calculus” by S.Roman, Bull. A.M.S., 13, 73-76(1985).

2. Review of ”Proof and Confirmation” by D. Bressoud, Amer. Math. Monthly, Feb. 2001.

3. Review of ”Kepler’s Problem” by G. Szpiro, Science 301, 1086 (Aug. 29, 2003).

4. Review of ”Mathematics by Experiments” by J. Borwein and D. Bailey, Amer. Sci., to appear.

Ph.D. Thesis “New Approaches and Results in the Theory of Discrete Analytic Functions”, Weizmann Institute of Science, Revovot. Israel, Aug. 1976. Advisor: Harry Dym.